What does a dimension of a homology group tell us? In particular, suppose we form an arbitrary simplicial complex $S(G)$ from a simple graph $G$. Then we compute the homology groups of $S(G)$ and note the ones which are nonzero. What useful information can we glean by computing the dimension of the nonzero homology groups?

See the Wikipedia article on Betti numbers. Could you please clarify what you mean by "suppose we form an arbitrary simplicial complex from a simple graph" - well if the simplicial complex is arbitrary we will get an arbitrary result, won't we?
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t.b.May 27 '11 at 17:33

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Is your question about homology in general or about the homology of the coloring complex in particular? If the former, perhaps the discussion at MO might be useful: mathoverflow.net/questions/640/…
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Qiaochu YuanMay 27 '11 at 18:39

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Homology is an incredibly rich and vast field, and homology groups can tell you all sorts of things about different spaces. Without more detail about what you're looking for, we can't really give you more information. Since you seem to have a background in combinatorics, I'd say this question is a bit like asking, "Suppose I form a vertex set and edges from a topological space. What useful information can we glean from the resulting graph?"
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MartianInvaderMay 27 '11 at 21:00

Why not try to answer the question yourself? If you want to start scratching the surface, compute the Euler characteristic of $S(G)$. This is readily done and expressible as a certain number that you can extract directly from the graph $G$. Is that number familiar to you?
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Ryan BudneyMay 27 '11 at 21:54

I see it is the alternating sum of the Betti Numbers.
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PEVMay 27 '11 at 22:07